736 
PEOEESSOE CAYLEY ON PEEP OTEN TI A L S . 
taken over the sphere x 2 . . . -j -z 2 =f 2 , 
a 2 c 2 e 2 
98. Second example. 0 the positive root of . • • +p+g+ y = 1 ’■> #+1 positive. 
Consider here the function 
V= f F'-'it+f*. . . t+h 2 )~idt; 
this satisfies the prepotential equation. We have in fact 
dV_ d6_' fV_ dH , (My 
da da ’ da 2 da 2 ® \^« y ’ 
rf 2 V ri 2 V 
with the like expressions for ; also 
Hence 
2q + l d\ 2q + 1 dd 
e de M e de’ 
□ V=-©D^-0'V^ 
or, substituting for and their values, this is 
Moreover V does not become infinite for any values of (a . . .c, e), e not =0 ; and it 
vanishes for points at oo ; and not only so, but for indefinitely large values of any of the 
coordinates (a. . . e, e) it reduces itself to a numerical multiple of ( a 2 . . . + c 2 +e 2 ) _ * s+? ; 
in fact in this case 0 is indefinitely large, =a 2 . . . + c 2 +e 2 : consequently throughout 
the integral t is indefinitely large, and we may therefore write 
that is 
V=r 
The conditions of the theorem are thus satisfied, and we have for § either of the 
formulae, 
?= 
m* + q) 
(e 2? W) 0 , g = 
~ r (jg+g) 
2 ( r i)T( 9 + i) 
(in the former of them q must be positive; in the latter it is sufficient if < 2+1 be 
positive). 
